In this paper, we propose the incrementally extensible folded hypercube (IEFH) graph as a new class of interconnection networks for an arbitrary number of nodes. We show that this system is optimal fault tolerant and almost regular (i.e., the difference between the maximum and the minimum degree of nodes is at most one). The diameter of this topology is half of that of the incomplete hypercube (IH), the supercube, or the IEH graph. We also devise a simple routing algorithm for the IEFH graph. Finally, we embed cycles and complete binary trees into this graph optimally.